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PC1221 Fundamentals of Physics I

Lectures 17 and 18Linear Momentum and Collisions

A/Prof Tay Seng Chuan

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Ground RulesGround Rules

Switch off your handphone and pager Be disciplined

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Linear Momentum

The linear momentumlinear momentum of a particle or an object that can be modeled as a particle of mass m moving with a velocity v is defined to be the product of the mass and velocity: p = m v

The terms momentum and linear momentum will be used interchangeably in this course, i.e., when we say momentum we also means linear momentum (which is in a straight line)

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Linear Momentum, cont Linear momentum is a vector

quantity Its direction is the same as the

direction of v The dimensions of

momentum (mass x velocity) are ML/T The SI units of momentum

are kg · m /s

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Linear Momentum, cont Momentum can be expressed in

component form (small letter p): px = m vx py = m vy pz = m vz

Momentum in x direction

Momentum in y direction

Momentum in z direction

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Newton and Momentum Newton called the product mvmv the

quantity of motionquantity of motion of the particle

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d md dm m

dt dt dt

vv pF a

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The time rate of change of the linear momentum of a particle is equal to the net force acting on the particle This is the form in which Newton

presented the Second Law It is a more general form than the one

we used previously This form also allows for mass

changes Momentum approach can be used to

analyse the motion in a system of particles

d md dm m

dt dt dt

vv pF a

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Conservation of Linear Momentum Whenever two or more particles in

an isolated system interact, the total momentum of the system remains constant The momentum of the system is

conserved, but the momentum of individual particle may not necessarily conserved.

The total momentum of an isolated system equals its initial momentum

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Conservation of Momentum, 2 Conservation of momentum can be

expressed mathematically in various ways ptotal = p1 + p2 = constant p1i + p2i= p1f + p2f

final Sum

Initial Sum

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Conservation of Momentum, 2 In component form for the various

directions, the total momentum in each direction is independentlyindependently conserved pix = pfx piy = pfy piz = pfz

Conservation of momentum can be applied to systems with any number of particles

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Conservation of Momentum, Archer Example The archer is

standing on a frictionless surface (ice). We know the mass of the archer (with bow) and the mass of the arrow, and the speed of the arrow. What will be the recoil speed of the archer?

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Conservation of Momentum, Archer Example Approaches to solve

this problem: Newton’s Second Law

– no, no information about F or a

Energy approach – no, no information about work or energy

Momentum – yes

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Let the system be the archer with bow (particle 1) and the arrow (particle 2)

There are no external forces in the x-direction, so it is isolated in terms of momentum in the x-direction

Total momentum before releasing the arrow is 0

The total momentum after releasing the arrow is

p1f + p2f = 0

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The archer will move in the opposite direction of the arrow after the release Agrees with Newton’s

Third Law Because the archer is

much more massive than the arrow, his acceleration and velocity will be much smaller than those of the arrow

p1f + p2f = 0, or,

m1v1f + m2v2f = 0

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Impulse and Momentum From Newton’s Second Law

Solving for dp (by cross multiplying) gives dp = Fdt By integration, we can find the change in momentum

over some time interval

The integral is called the impulseimpulse (I )of the force F acting on an object over the time t

The impulse imparted to a particle by a force is equal to the change in the momentum of the particle (impulse-momentum theorem). This is equivalent to Newton’s Second Law.

f

i

t

f i tdt p p p F I

= 1

F

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Impulse is a vector quantity

The magnitude of the impulse is equal to the area under the force-time curve

Dimensions of impulse are M (L T-2) T = M L T-1 = M L / T ([] signs removed for simplicity)

Impulse is not a property of the particle, but a measure of the change in momentum of the particle

f

i

t

f i tdt p p p F I

Force

Time

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The Fun FairThe Fun Fair

You have paid $10 to a stall owner at a fun fair. You have to knock down 3 pins arranged in a triangle from a distance. If you can do it you will be given a teddy bear. The stall owner gives you two items before you proceed:

(i) rounded bean bag

(ii) rubber ball

Both of them have same mass and same radius. You are allowed to use only one of the items to knock the pins down. Which item should you use? Why?

Think carefully and you give the answer when we meet.

Answer:

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Air cushion is used by firemen to save lives. How does it work?

1 October 2008

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If I throw an egg at you, how are you If I throw an egg at you, how are you going to catch it without messing out going to catch it without messing out yourself? yourself?